1.6 Trig Functions

1. 1.6 Trig Functions

2. 1.6 Trig Functions The Mean Streak, Cedar Point Amusement Park, Sandusky, OH

3. P positive angle x O x O negative angle P Trigonometry Review (I) Introduction By convention, angles are measured from the initial line or the x-axis with respect to the origin. If OP is rotated counter-clockwise from the x-axis, the angle so formed is positive. But if OP is rotated clockwise from the x-axis, the angle so formed is negative.

4. r r 1c r (II) Degrees & Radians Angles are measured in degrees or radians. Given a circle with radius r, the angle subtended by an arc of length r measures 1 radian. • Care with calculator! Make sure your calculator is set to radians when you are making radian calculations.

5. Note: (III) Definition of trigonometric ratios y P(x, y) r y  x x Do not write cos-1q, tan-1q .

6. All +ve sin +ve 1st 2nd 3rd 4th tan +ve cos +ve From the above definitions, the signs of sin , cos & tan in different quadrantscan be obtained. These are represented in the following diagram:

7. (IV) Trigonometrical ratios of special angles What are special angles? 30o, 45o, 60o, 90o, … Trigonometrical ratios of these angles are worth exploring

8. 1 -1 0 p 2p sin 360° = 0 sin 180° = 0 sin 0° = 0 sin 90° = 1 sin 270° = -1

9. 1 -1 0 p 2p cos 360° = 1 cos 180° = -1 cos 0° = 1 cos 270° = 0 cos 90° = 0

10. p 0 2p tan 360° = 0 tan 180° = 0 tan 0° = 0 tan 270° is undefined tan 90° is undefined

11. Using the equilateral triangle (of side length 2 units) shown on the right, the following exact values can be found.

12. Complete the table. What do you observe?

14. or 2p+q 1st quadrant 2nd quadrant 3rd quadrant Important properties:

15. 4th quadrant Important properties: or 2p+q or-q In the diagram, q is acute. However, these relationships are true for all sizes of q.

16. Two angles that sum up to 90° or radians are called complementary angles. are complementary angles. Complementary angles E.g.: 30° & 60° are complementary angles. Recall:

17. Principal range Principal Angle & Principal Range Example: sinθ = 0.5 Restricting y= sinθ inside the principal range makes it a one-one function, i.e. so that a unique θ= sin-1y exists

18. Since sin is positive, it is in the 1st or 2nd quadrant Basic angle, α = Therefore Hence, Example:sin . Solve for θ if

19. P(x, y) r y A x O Since and , (VI) 3 Important Identities By Pythagoras’ Theorem, Note: sin 2A= (sin A)2 cos 2A= (cos A)2 sin2A+ cos2A= 1

20. (1) sin2A + cos2A= 1 (2) tan2 A +1 = sec2 A (3) 1 + cot2 A= csc2 A (VI) 3 Important Identities Dividing (1) throughout by cos2A, tan 2x = (tan x)2 Dividing (1) throughout by sin2A,

21. (VII) Important Formulae (1) Compound Angle Formulae

22. E.g. 4:It is given that tan A = 3. Find, without using calculator, • (i) the exact value of tan , given that tan (+A) = 5; • the exact value of tan , given thatsin(+A) = 2 cos ( –A) Solution: (i) Given tan (+A) = 5 and tan A= 3,

23. Proof: (i) sin 2A = 2 sin A cos A (iii) (2) Double Angle Formulae (ii) cos 2A = cos2A– sin2A = 2 cos2A– 1 = 1 – 2 sin2A

24. When you use trig functions in calculus, you must use radian measure for the angles. The best plan is to set the calculator mode to radians and use when you need to use degrees. o 2nd Trigonometric functions are used extensively in calculus. If you want to brush up on trig functions, they are graphed on page 41.

25. Cosine is an even function because: Even and Odd Trig Functions: “Even” functions behave like polynomials with even exponents, in that when you change the sign of x, the y value doesn’t change. Secant is also an even function, because it is the reciprocal of cosine. Even functions are symmetric about the y - axis.

26. Sine is an odd function because: Even and Odd Trig Functions: “Odd” functions behave like polynomials with odd exponents, in that when you change the sign of x, the sign of the y value also changes. Cosecant, tangent and cotangent are also odd, because their formulas contain the sine function. Odd functions have origin symmetry.

27. is a stretch. is a shrink. The rules for shifting, stretching, shrinking, and reflecting the graph of a function apply to trigonometric functions. Vertical stretch or shrink; reflection about x-axis Vertical shift Positive d moves up. Horizontal shift Horizontal stretch or shrink; reflection about y-axis Positive c moves left. The horizontal changes happen in the opposite direction to what you might expect.

28. is the amplitude. is the period. B A C D When we apply these rules to sine and cosine, we use some different terms. Vertical shift Horizontal shift

29. The sine equation is built into the TI-89 as a sinusoidal regression equation. For practice, we will find the sinusoidal equation for the tuning fork data on page 45. To save time, we will use only five points instead of all the data.

30. ENTER ENTER STO Tuning Fork Data Time: .00108 .00198 .00289 .00379 .00471 Pressure: .200 .771 -.309 .480 .581 .00108,.00198,.00289,.00379,.00471 2nd { 2nd } alpha L 1 6 3 9 alpha L 1 , alpha L 2 2nd MATH SinReg The calculator should return: Statistics Regressions Done

31. ENTER ENTER 6 3 9 alpha L 1 , alpha L 2 2nd MATH SinReg The calculator should return: Statistics Regressions Done 6 8 2nd MATH Statistics ShowStat The calculator gives you an equation and constants:

32. WINDOW ENTER ENTER Y= We can use the calculator to plot the new curve along with the original points: x y1=regeq(x) ) regeq 2nd VAR-LINK Plot 1

33. WINDOW GRAPH ENTER ENTER Plot 1

34. WINDOW GRAPH You could use the “trace” function to investigate the pressure at any given time.

35. Trig functions are not one-to-one. However, the domain can be restricted for trig functions to make them one-to-one. These restricted trig functions have inverses. Inverse trig functions and their restricted domains and ranges are defined on page 47. p